The leading order effects of the filament response to a viscoelastic fluid are encoded in the variable \(\xi \) . As seen from equation (REF ), \(\xi \)  modifies the evolution of \(\mathbf {X}\)  in the same way as the preferred curvature \(\kappa _0\) , and, from equation (), may be interpreted as the “memory” of the curvature difference \(\kappa -\kappa _0\)  at previous times.
The parameters \(\delta \ge 0\)  and \(\mu \ge 0\)  are associated with viscoelasticity: \(\delta \)  is the additional relaxation time of the filament due to viscoelastic effects of the fluid, and \(\mu \)  relates to the additional (polymer) viscosity of the viscoelastic medium.
Note that if \(\mu =0\)  or if \(\delta =0\) , we recover the classical Newtonian formulation (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]})
\(\frac{\partial \mathbf {X}}{\partial t} = -({\bf I}+\gamma \mathbf {X}_s\mathbf {X}_s^{\rm T})\big (\mathbf {X}_{sss}-\tau \mathbf {X}_s-(\kappa _0)_s\mathbf {e}_{\rm n}\big )_s\) 
